An object can be seen if it reflects, scatters or radiates light. This light can be described by a set of waves of an electromagnetic field. If the same electromagnetic field, called hereinafter in this specification, the model field, were to be recreated by other means, an observer would see the same image. The simplest way of creating a copy of a model field is by means of a photograph. This method provides a copy of the field intensity distribution in the lens focal plane, but does not save the field phase distribution. Phase distribution contains information about location in object space. That is why a true image can be seen only from one viewpoint.
A method of preserving this phase information in the copy of the model field is by means of a hologram. The first holograms were formed by Gabor in research on reducing the aberrations in electron microscopes. A simple hologram can be produced by illuminating the object with coherent light, and using the same light as a reference field to produce an interference pattern inside the photographic recording material. Under the appropriate conditions, the resulting photograph will memorize both the intensity and phase information of the model field. In conventional holography, this information is read using the same beam of coherent light as was used for the reference light.
Let U({overscore (r)})=u({overscore (r)})ei(u({overscore (r)})−ax) represent the field reflected or scattered by the object, the model field, and                let U0({overscore (r)})=u0ei({overscore (k)}o{overscore (r)}−ax) represent the plane reference field, where:        F is the radius vector,        {overscore (k)}0 is the wave vector,        ω is the frequency,        v(F) is the phase, and        u(F) is the amplitude of the model field.The resulting field on the surface of the photograph is given by:Φ(x,y)=u(x,y)ei(v(x,y)−ax) +u0eI(kny,v−ax) and the average intensity of field is given by:I(x,y)=u(x,y)2+(u0)2+u(x,y)u0(eI(v(x,y)−kax x+knyy )+e−1(v(u(x,y)+kaxx+kny,y) ) where:        x and y are the co-ordinates of the point on the two-dimensional photograph where the field is calculated,        u(x,y) is the projection of u(F) onto the plane of the photograph, and        kx and ky are the projections of the wave vector onto the x- and y-axes.        
Assuming that the transparency of the resulting holographic photograph is proportional to the square of the exposure. When illuminated by the reference field U0(F), three waves result, as is well known in the art. The first is a simple forward scattered wave, the modulated reference field. The second is a virtual image, normal to the surface of the photo. The third is a copy of the required model field.
Mathematically, the resulting holographic photograph of the interference pattern can be considered to be a Fourier representation of the model field. The model field is generated by interference of the reference light beam with the hologram. The resulting image is produced in practice, by light scattered from a predefined set of tiny crystals within the hologram photographic recording material, each with different size and density.
Holographic images are three dimensional and remarkably realistic. The original uses of holography in imaging and display applications has now grown to include applications such as optical, RF and acoustic filtering, holographic image processing, holographic or diffractive optical elements including lenses, aspherical optics, beam splitters and aberration correctors, night vision devices, helmet displays, memory devices, phase contrast microscopy, optical disc readouts and many others.
The problem with the prior art conventional method of producing holograms on photographic recording materials, is that it requires the use of a complex optical set-up, with coherent writing and reading beams for respectively producing and displaying the holograms. A number of methods and apparatus have been described in the prior art for producing holograms using light-sensitive materials. For example, U.S. Pat. No. 5,291,317 on “Holographic diffraction grating patterns and methods for creating the same” to C. Newswanger describes methods and apparatus for creating a plurality of holographic diffraction grating patterns in a raster scan fashion. U.S. Pat. No. 5,592,313 on “Methods and apparatus for making holograms including a variable beam splitter assembly” and U.S. Pat. No. 5,745,267 on “Apparatus for making holograms”, to S. J. Hart describe methods for producing composite holograms using substantially planar, photo-sensitive substrates. A real-time holography system using a CCD camera as the photo-sensitive recording medium, is described by N. Hashimoto in U.S. Pat. No. 5,515,133 “Real time holography system”.
Furthermore, such holograph production methods require either the presence of the object sample itself or involve a complicated method of wave construction for synthetically reconstructing the object beam. This wave reconstruction could include beam splitting, amplitude transformation, phase processing, followed by recombination of the reconstructed object beam with the reference beam to form the interference pattern. This has limited the use of holograms, because of the complexity and high precision of the optical equipment required to produce them. There is therefore a widespread need for a novel method of producing holograms in transparent materials, without the need for complicated optical interference arrangements, thereby reducing the complexity and cost of production of such holograms.
Computer generated holograms, hereinafter CGH's, have been described by B. R. Brown and A. W. Lohmnann in the article “Complex Spatial Filtering with Binary Masks”, published in Applied Optics, Vol. 5, p. 967ff, (1966). The CGH's described therein were used for optical spatial filtering. The hologram is represented by assigning field values to discrete pixels. The number of values is dependent on the size of the image and its resolution. Pixel values can be calculated by means of scalar diffraction theory and form a Fourier transform of the model field. In evaluating the Fourier series, some approximations are necessary. One of these approximations, known as the Detour phase error, has given its name to a common CGH variant. Other approaches have also been developed and the resulting holograms known as Nondetour Holograrns. Some such CGH's are described in the article entitled “Binary Synthetic Holograms” by W. H. Lee published in Applied Optics, Vol. 13, p. 1677ff, (1974).
Three main features of a CGH should be emphasized:    (a) neither object nor reference field need really exist, such that any convenient mathematical representation of the object and any reference wave front can be utilized;    (b) the calculation is essentially the evaluation of the inverse diffraction effect, in that the diffracted field is given and the pixel values of the holograph need to be calculated;    (c) the mathematical nature of a CGH enables the construction of idealized images and filters, by means of compensation for such effects as beam aberrations, system noise, beam divergence, beam phase shift, and others.
A number of different artifacts and aliases are produced as a result of the discrete nature of defining the value assigned to each pixel, such as because of the finite number of pixel positions, their coherent positioning and the quantization of the pixel values. Such effects are well documented in the prior art, in a number of publications, such as “The Effect of Finite Sampling in Holography”, by W. T. Cathey, in Optik, Vol. 27, p. 317ff, (1968); “Aliasing Error in Digital Holography” by J.Buklew and N. C. Galaher, in Journal of Applied Optics. Vol. 15, p. 2183ff, (1976); “Some Effects of Fourier Domain Phase Quantization” by J. W. Goodman and A. M. Silvestri, in IBM Journal of Research and Development, Vol. 14, p. 478ff, (1970); and “Quantization Noise in Binary Holograms” by P. S. Naidu, in Optics Communications, Vol. 15, p. 361ff, (1975).
Specific mathematical techniques have been described, for reducing these effects, such as those described by B. R. Brown and A. W. Lohmann in “Computer Generated Binary Holograms”, published in the IBM Journal of Research and Development, Vol. 13, p.168ff, (1969), and by J.Buklew and N. C. Galaher, Jr., in “Comprehensive Error Models and a Comparative Study of Some Detour Phase Holograms”, published in Applied Optics, Vol. 18, p.286lff, (1979).
The main lirmitation of prior art CGH's is that they can only be constructed using two dimensional recording techniques. such as surface chemical etching, lithography, ion etching, electron beam processing and silk screen printing. Hence such a CGH represents only a two dimensional slice of a complete volumetric interference pattern. As has been shown by Yu. N. Denisyuk in the article “Optical Properties of an Object as Mirrored in the Wave Field of its Scattered Radiation”, published in Optics and Spectroscopy, Vol. 15, pp. 522-532, (1963), unlike the case of three-dimensional volume holographs, for such a 2-D holograph, this means that there is no difference between direct and inverse wave propagation, that a virtual image always exists, and that there is no wavelength selectivity. Consequently, such a CGH is limited and can produce only a monochrome image or holographic optical element, HOE.
There therefore exists a serious need for a method of production of computer generated volume holograms, which overcome the disadvantages and drawbacks of prior art holograms.
The disclosures of all publications mentioned in this section and in the other sections of the specification, and the disclosures of all documents cited in the above publications, are hereby incorporated by reference.